for all \(x,y\) in \(G\text{.}\) A homomorphism that is both injective (one-to-one) and surjective (onto) is called an isomorphism of groups. If \(\phi\colon G\to H\) is an isomorphism, we say that \(G\) is isomorphic to \(H\text{,}\) and we write \(G\approx H\text{.}\)

Proposition2.4.5.Basic properties of homomorphisms.

Let \(\phi\colon G\to H\) be a homomorphism of groups. Let \(e_G,e_H\) denote the identity elements of \(G,H\text{,}\) respectively. We have the following.

(identity goes to identity) \(\phi(e_G) = e_H\)

(inverses go to inverses) \(\phi\left(g^{-1}\right) = \left(\phi(g)\right)^{-1}\) for all \(g\in G\)

\(\ker(\phi)\) is a subgroup of \(G\)

\(\phi(G)\) is a subgroup of \(H\)

(preimage sets are cosets of the kernel) \(\phi(x)=y\) if and only if \(\phi^{-1}(y) =
x\ker(\phi)\)

\(\phi(a)=\phi(b)\) if and only if \(a\ker(\phi)=b\ker(\phi)\)

\(\phi\) is one-to-one if and only if \(\ker(\phi)=\{e_G\}\)

Proposition2.4.6.\(G/K\) is a group if and only if \(K\) is a kernel.

Let \(K\) be a subgroup of a group \(G\text{.}\) The set \(G/K\) of cosets of \(K\) forms a group, called a quotient group (or factor group), under the operation

if and only if \(K\) is the kernel of a homomorphism \(G\to G'\) for some group \(G'\text{.}\)

Corollary2.4.7.(First Isomorphism Theorem).

Let \(\phi\colon G\to H\) be a homomorphism of groups. Then \(G/\ker(\phi)\) is isomorphic to \(\phi(G)\) via the map \(g\ker(\phi) \to \phi(g)\text{.}\)

Definition2.4.8.Normal subgroup.

A subgroup \(H\) of a group \(G\) is called normal if \(ghg^{-1}\in H\) for every \(g\in G\text{,}\)\(h\in H\text{.}\) We write \(H\trianglelefteq G\) to indicate that \(H\) is a normal subgroup of \(G\text{.}\)

Proposition2.4.9.Characterization of normal subgroups.

Let \(K\) be a subgroup of a group \(G\text{.}\) The following are equivalent.

\(K\) is the kernel of some group homomorphism \(\phi\colon G\to H\)

\(G/K\) is a group with multiplication given by Equation (2.4.1)

First, suppose \(K=\ker(\phi)\) for some homomorphism \(\phi\colon G \to G'\text{.}\) Explain why Item 6 of Proposition 2.4.5 can be rephrased to say that there is a one-to-one correspondence \(G/K
\leftrightarrow \phi(G)\) given by \(gK\leftrightarrow
\phi(g)\text{.}\) Now use the bijection \(G/K\leftrightarrow
\phi(G)\) to impose the group structure of \(\phi(G)\) (Item 4 of Proposition 2.4.5) on \(G/K\text{.}\) Conversely, if \(G/K\) is a group with the group operation (2.4.1), define \(\phi\colon G\to
G/K\) by \(\phi(g)=gK\text{,}\) then check that \(\phi\) is a homomorphism and that \(\ker(\phi)=K\text{.}\)

Use the fact that \(\gcd(m,n)\) is the least positive integer of the form \(sm+tn\) over all integers \(s,t\) (see Exercise 2.3.2.4). Use this to solve \(ax=1 \pmod{n}\) when \(a,n\) are relatively prime.

8.Another construction of \(\Z_n\).

Let \(n\geq 1\) be an integer and let \(\omega=e^{i2\pi/n}\text{.}\) Let \(\phi\colon
\Z\to S^1\) be given by \(k\to \omega^k\text{.}\)

Show that the the image of \(\phi\) is the group \(C_n\) of \(n\)th roots of unity.

Show that \(\phi\) is a homomorphism, and that the kernel of \(\phi\) is the set \(n\Z=\{nk\colon k\in \Z\}\text{.}\)

Conclude that \(\Z/\!(n\Z)\) is isomorphic to the group of \(n\)-th roots of unity.

9.Isomorphic images of generators are generators.

Let \(S\) be a subset of a group \(G\text{.}\) Let \(\phi\colon G\to H\) be an isomorphism of groups, and let \(\phi(S)=\{\phi(s)\colon s\in S\}\text{.}\) Show that \(\phi(\langle S\rangle)=\langle \phi(S)\rangle\text{.}\)

10.Conjugation.

Let \(G\) be a group, let \(a\) be an element of \(G\text{,}\) and let \(C_a\colon G\to G\) be given by \(C_a(g)=aga^{-1}\text{.}\) The map \(C_a\) is called conjugation by the element \(a\) and the elements \(g,aga^{-1}\) are said to be conjugate to one another.

Show that \(C_a\) is an isomorphism of \(G\) with itself.

Show that "is conjugate to" is an equivalence relation. That is, consider the relation on \(G\) given by \(x\sim y\) if \(y=C_a(x)\) for some \(a\text{.}\) Show that this is an equivalence relation.

11.Isomorphism induces an equivalence relation.

Prove that "is isomorphic to" is an equivalence relation on groups. That is, consider the relation \(\approx\) on the set of all groups, given by \(G\approx H\) if there exists a group isomorphism \(\phi\colon G\to H\text{.}\) Show that this is an equivalence relation.

Show that Item 3 implies Item 2. The messy part of this proof is to show that multiplication of cosets is well-defined. This means you start by supposing that \(xK=x'K\) and \(yK=y'K\text{,}\) then show that \(xyK=x'y'K\text{.}\)

14.Further characterizations of normal subgroups.

Show that Item 3 is equivalent to the following conditions.

\(gKg^{-1}= K\) for all \(g\in G\)

\(gK = Kg\) for all \(g\in G\)

15.Automorphisms.

Let \(G\) be a group. An automorphism of \(G\) is an isomorphism from \(G\) to itself. The set of all automorphisms of \(G\) is denoted \(\Aut(G)\).

Show that \(\Aut(G)\) is a group under the operation of function composition.